Other ways to compute the torsion subgroup of elliptic curves Suppose I have a family of elliptic curves $E_{n}/\mathbb{Q}$. I would like to determine the torsion subgroup of $E_{n}(\mathbb{Q})$ denoted by $E_{n}(\mathbb{Q})_{\textrm{tors}}$. Two ways to do this are using Nagell-Lutz and computing the number of points over $\mathbb{F}_{\ell}$ for various $\ell$. Are there other ways to determine the torsion subgroup of an elliptic curve?
 A: For a fixed curve over $\mathbb Q$, the easiest way is to check Cremona's tables (!), since it is pretty unlikely that your curve has conductor big enough not to be there.  
Sorry for the cheeky answer; here is another slightly more serious one:
I think that using the methods you suggest is pretty
standard; as Don Antonio mentions, Mazur's theorem also gives a pretty
good absolute upper bound.  
One alternative to actually working mod $\mathbb F_{\ell}$ for various $\ell$
is to compute that modular form attached to $E$ (or in practice, look it
up in a table).  Then it is pretty easy to look for congruences
$a_{\ell} \equiv 1 + \ell mod p$ and hence at least determine the possible primes that divide the order of the torsion subgroup.
In the end, if you are using a table, as I said before this table will also likely just contain a precise description of the torsion subgroup.  Still, 
the idea of relating the structure of the torsion to congruences between
the modular form of $E$ an an Eisenstein series is important.  (E.g. it
is the basic mechanism in Mazur's proof of this theorem.)
One thing to remember though is that the modular form, or equivalently
the number of $\mathbb F_{\ell}$ points, is an isogeny invariant, so
that even if computing $\mathbb F_{\ell}$-points suggests a $p$-torsion
point, there may not actually be such a point on your curve (but I guess
there will be on some $p$-isogenous curve). (You can see an example
in the isogeny class of $X_0(11)$, with $p = 5$.)
